Assessment and Future Directions of Nonlinear Model Predictive ...

554 S. Gros et al.Ṡ = −H xx + S(F u Huu −1 H ux − F x ) (9)+(H xu Huu −1 FuT − Fx T )S + SF u Huu −1 Fu T S + H xu Huu −1 H uxS(t k + T )=Φ(x(t k + T )) (10)The S matrix is computed backward in time. This computation can be numericallydemanding. Controller (7)-(10) is termed the NE-controller. Note that itdoes not take constraints into account.3 Two-Time-Scale Control SchemeThe repeated solution of (2)-(4) provides feedback to the system. Yet, sincethe time necessary to perform the optimization can be rather large comparedto the system dynamics, the feedback provided by the re-optimization tends tobe too slow to guarantee performance and robustness. Hence, it is proposed toadd a fast feedback loop in the form of a NE-controller. The resulting controlscheme is displayed in Figure 1. The NE-controller operates in the fast loop at asampling frequency appropriate for the system, while the reference trajectoriesare generated in the slow loop at a frequency permitting their computation. Notethat, if the time-scale separation between the two loops is sufficient, u ref (t) canbe considered as a feedforward term for the fast loop.3.1 Trajectory GenerationThe generation of the reference trajectories u ref (t) andx ref (t) can be computedvia optimization (e.g. nonlinear MPC) or direct system inversion (as is possiblefor example for flat systems [10]).3.2 Tracking NE-ControllerThe NE-controller (7)-(10) can be numerically difficult to compute. Its computationis simplified if the optimization problem considers trajectory tracking.Indeed, for tracking the trajectories u ref (t) andx ref (t), Φ and L can bechosen as:Φ = 1 2 (x − x ref) T P (x − x ref ) (11)L = 1 2 (x − x ref) T Q(x − x ref )+ 1 2 (u − u ref) T R(u − u ref ) (12)for which the solution to problem (2)-(3) is u ∗ (t) =u ref (t) andx ∗ (t) =x ref (t).Furthermore, the adjoints read:˙λ = −Hx T = −Fx T λ − Q(x − x ref ) (13)λ(t k + T )=Φ x (t k + T ) = 0 (14)